Helin Guo, Yimin Zhang, Huan-Song Zhou
We study a generalized Kirchhoff type equation with trapping potential. The existence and blow-up behavior of solutions with normalized L2-norm for this problem are discussed.
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In this paper we will apply the modified potential well method and variational method to the study of the long time behaviors of solutions to a class of parabolic equation of Kirchhoff type. Global existence and blow up in finite time of solutions will be obtained for arbitrary initial energy. To be a little more precise, we will give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. The decay rate of the $L^2(Ω)$ norm is also obtained for global solutions in these cases. Moreover, some suffici…
In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. We use the potential well method to give a threshold result of global existence and finite-time blow-up for the weak solutions with initial energy $J(u_0)\leq d$. When the initial energy $J(u_0)>d$, we find another criterion for the vanishing solution and blow-up solution. We also get the exponential decay rate of the global solution and life span of the blow-up solution. Meanwhile, we study the corresponding stationary problem and establish a convergence relat…
Billiards, Refraction, and Blow-Up for Kirchhoff Equations
本文通过几何光学中的椭圆台球折射反射机制构造双模态Kirchhoff系统的横向异宿轨道，结合路线图定理，证明受迫抽象Kirchhoff方程存在有限时间爆破解。
Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities
本文研究了大型光滑星形区域及全空间上具有混合非线性项的 Kirchhoff 方程的给定质量解，分析了其存在性、多重性与渐近行为，并指出势函数的存在导致标准 Pohozaev 恒等式方法失效。
It is well-known that the classical hyperbolic Kirchhoff equation admits infinitely many simple modes, namely time-periodic solutions with only one Fourier component in the space variables. In this paper we assume that, for a suitable choice of the nonlinearity, there exists a heteroclinic connection between two simple modes with different frequencies. Under this assumption, we cook up a forced Kirchhoff equation that admits a solution that blows-up in finite time, despite the regularity and boundedness of the forcing term. The forcing term can be chosen with the maximal regularity that preven…
Our aim in this article is to study a nonlinear viscoelastic Kirchhoff equation with strong damping, Balakrishnan-Taylor damping, nonlinear source and dynamical boundary condition. Firstly, we prove the local existence of solutions by using the Faedo-Galerkin approximation method combined with a contraction mapping theorem. We then prove that if the initial data enter into the stable set, the solution globally exists, and if the initial data enter into the unstable set, the solution blows up in a finite time. Moreover, we obtain a general decay result of the energy, from which the usual expone…
Long-time behavior for the Kirchhoff diffusion problem with magnetic fractional Laplace operator
研究磁分数拉普拉斯算子驱动的Kirchhoff扩散问题，证明其不会发生无限时间爆破，方法为势井方法结合能量与Nehari泛函。
In this paper, a parabolic type Kirchhoff equation and its stationary counterpart are considered. For the evolution problem, the precise decay rates of the weak solution and of the corresponding energy functional are derived. For the stationary problem, a ground-state solution is obtained by applying Lagrange multiplier method. Moreover, the asymptotic behaviors of the general global solutions are also described. These results extend some recent ones obtained in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers …
We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.
In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_Ω|\nabla u|^2 \dif x +b\right)Δu = λu+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $λ\in \mathbb{R}$ and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of $a$, $b$, $λ$ and the initial data for the following cases: (i) $1<p<3$, (ii) $p=3$ and $a>1/Λ$, (iii) $p=3$, $a \leq 1/Λ$ and $\lam <b\lam_1$, (iv) $p=3$, $a < 1/Λ$ and $\lam >b\lam_1$, (v) $p>3$…
In present paper, we study the normalized solutions $(λ_c, u_c)\in \R\times H^1(\R^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\R^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3 $$ satisfying the normalization constraint $ \displaystyle\int_{\R^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c…
In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure defined on the interval. Under Dirichlet boundary conditions we establish the exclusion of boundary blowup points, we show that the concentration mass does not have residual L1-terms and we determine the location of blowup points in terms of Kirchhoff's Hamiltonian. We …
Propagation of Singularity with Normally Hyperbolic Trapping
该研究通过构造基于不稳定流形与纤维无穷远交集吹胀的新型符号类，证明了带法向双曲捕获的微局部估计，得出Kerr(-de Sitter)时空标量波动方程的该类估计较标准奇性传播的额外损失任意小的结论。
Normalized solutions of $L^2$-supercritical Kirchhoff equations in bounded domains
本文在有界区域中研究 L^2 超临界 Kirchhoff 方程的归一化解，利用带 Morse 指标信息的参数化极小极大方法和爆破分析建立山路型解的存在性，并分析 b→0 时的渐近行为。
Consider the following eigenvalue problem of p-Laplacian equation \begin{equation}\label{P} -Δ_{p}u+V(x)|u|^{p-2}u=μ|u|^{p-2}u+a| u|^{s-2}u, x\in \mathbb{R}^{n}, \tag{P} \end{equation} where $a\geq0$, $p\in (1,n)$ and $μ\in\mathbb{R}$. $V(x)$ is a trapping type potential, e.g., $\inf\limits_{x \in \mathbb{R}^n}V(x)< \lim\limits_{|x|\rightarrow+\infty}V(x)$. By using constrained variational methods, we proved that there is $a^*>0$, which can be given explicitly, such that problem (\ref{P}) has a ground state $u$ with $\|u\|_{L^p}=1$ for some $μ\in \mathbb{R}$ and all $a\in [0,a^*)$, but (\ref{P…
Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior
本文研究了带捕获势的耦合伪相对论Hartree系统的基态，通过约束最小化问题完全分类了基态存在性，并精确分析了耦合系数趋于临界值时极小子的爆破与集中行为及最优爆破速率。
We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones …
We study the existence and blow-up behavior of minimizers for $E(b)=\inf\Big\{\mathcal{E}_b(u) \,|\, u\in H^1(R^2), \|u\|_{L^2}=1\Big\},$ here $\mathcal{E}_b(u)$ is the Kirchhoff energy functional defined by $\mathcal{E}_b(u)= \int_{R^2} |\nabla u|^2 dx+ b(\int_{R^2} |\nabla u|^2d x)^2+\int_{R^2} V(x) |u(x)|^2 dx - \frac{a}{2} \int_{R^2} |u|^4 dx,$ where $a>0$ and $b>0$ are constants. When $V(x)= -|x|^{-p}$ with $0<p<2$, we prove that the problem has (at least) a minimizer that is non-negative and radially symmetric decreasing. For $a\ge a^*$ (where $a^*$ is the optimal constant in the Gagliar…
Global and local existence of solutions for a novel type of parabolic Kirchhoff system with singular term
本文研究了新型抛物型Kirchhoff系统中解的全局和局部存在性，通过Faedo-Galerkin方法证明弱解存在，并探讨了初始能量水平对解爆破和稳定性的不同影响。
The uniqueness and concentration behavior of solutions for a nonlinear fractional Schrödinger system
本文研究了二维耦合分数阶薛定谔方程组的解的唯一性及集中行为，通过L^2约束极小化问题和隐函数定理证明了解的唯一性，并分析了在临界值附近解的集中现象及爆破速率。
The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in $\mathbb{R}^2$, solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under certain type of trapping potentials $V_i(x)$ ($i=1,2$), the existence, non…
Concentration behavior of normalized ground states for mass critical Kirchhoff equations in bounded domains
研究质量临界Kirchhoff方程在有界域中归一化基态的集中行为，分析参数a趋近于0时极小值的极限特性及局部唯一性。
We consider a 3D quantum system of $N$ identical bosons in a trapping potential $|x|^p$, with $p\geq0$, interacting via a Newton potential with an attractive interaction strength $a_{N}$. For a fixed large $N$ and the coupling constant $a_{N}$ smaller than a critical value $a_{*}$ (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground state energy as well as the ground states when $a_{N}$ approaches $a_{*}$ sufficiently slowly in the limit $N\to\infty$. The blow-up profile is given by the Gagliardo-Nire…
We consider a Gross-Pitaevskii equation which appears as a model in the description of dipolar Bose-Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scatteri…
On a structural acoustic model with logarithmic supercritical source terms
研究由三维有界域上的半线性波动方程与边界上的Kirchhoff-Love板方程耦合的结构声学模型，通过施加非线性阻尼和对数型超临界源项，分析局部和全局适定性、势阱解的能量衰减率以及解的爆破。
A McKean--Vlasov equation with positive feedback and blow-ups
研究具有正反馈的粒子系统平均场模型导出的McKean--Vlasov方程，通过不动点论证构造可微分解，并在càdlàg函数类中证明唯一性，扩展初始条件并引入新技术以支持混沌传播结果。
We study minimizers of the pseudo-relativistic Hartree functional $$\mathcal{E}_{a}(u):=\|(-Δ+m^{2})^{1/4}u\|_{L^{2}}^{2}-\frac{a}{2}\int_{\mathbb{R}^{3}}(\left|\cdot\right|^{-1}\star |u|^{2})(x)|u(x)|^{2}{\rm d}x+\int_{\mathbb{R}^{3}}V(x)|u(x)|^{2}{\rm d}x$$ under the mass constraint $\int_{\mathbb{R}^3}|u(x)|^2{\rm d}x=1$. Here $m>0$ is the mass of particles and $V\geq 0$ is an external potential. We prove that minimizers exist if and only if $a$ satisfies $0\leq a<a^{*}$, and there is no minimizer if $a\geq a^*$, where $a^*$ is called the Chandrasekhar limit}. When $a$ approaches $a^*$ from…
This paper is concerned with global existence as well as infinite-time blowups of classical solutions to the following fully parabolic kinetic system \begin{equation} \begin{cases} u_t=Δ(γ(v)u) v_t-Δv+v=u \end{cases} \qquad(0.1)\end{equation} in a smooth bounded domain $Ω\subset\mathbb{R}^n$, $n\geq1$ with no-flux boundary conditions. This model was recently proposed in [8,20] to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function $γ(\cdot)$, which is decreasing in $v$ and will vanish as $v$ tends…
We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schrödinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points. Precisely, applying the finite dimensional reduction method, we first obtain the existence of $k$-peak concentrated solutions and especially describe the relationship between the chemical potential $μ$ and the attractive interaction $a$. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the $k$…
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